Quantum computational algorithms exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms1,2,3. Shor's quantum algorithm4 for fast number factoring is a key example and the prime motivator in the international effort to realize a quantum computer5. However, due to the substantial resource requirement, to date there have been only four small-scale demonstrations6,7,8,9. Here, we address this resource demand and demonstrate a scalable version of Shor's algorithm in which the n-qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one-third of that required in the standard protocol10,11. Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N = 21. The algorithmic output is distinguishable from noise, in contrast to previous demonstrations. These results point to larger-scale implementations of Shor's algorithm by harnessing scalable resource reductions applicable to all physical architectures.
This is a preview of subscription content, access via your institution
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).
Deutsch, D. Quantum theory, the Church–Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A 400, 97–117 (1985).
Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).
Shor, P. W. in Proceedings of the 35th Annual Symposium on Foundations of Computer Science (ed. Goldwasser, S.) 124–134 (IEEE Computer Society Press, 1994).
Ladd, T. D. et al. Quantum computers. Nature 464, 45–53 (2010).
Vandersypen, L. M. K. et al. Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–887 (2001).
Lu, C-Y., Browne, D. E., Yang, T. & Pan, J-W. Demonstration of a compiled version of Shor's quantum factoring algorithm using photonic qubits. Phys. Rev. Lett. 99, 250504 (2007).
Lanyon, B. P. et al. Experimental demonstration of a compiled version of Shor's algorithm with quantum entanglement. Phys. Rev. Lett. 99, 250505 (2007).
Politi, A., Matthews, J. C. F. & O'Brien, J. L. Shor's quantum factoring algorithm on a photonic chip. Science 325, 1221 (2009).
Parker, S. & Plenio, M. B. Efficient factorization with a single pure qubit and logN mixed qubits. Phys. Rev. Lett. 85, 3049–3052 (2000).
Mosca, M. & Ekert, A. in Lecture Notes in Computer Science Vol. 1509, 174–188 (Springer, 1999).
Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).
Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nature Chem. 2, 106–111 (2010).
Veis, L. & Pittner, J. Quantum computing applied to calculations of molecular energies: CH2 benchmark. J. Chem. Phys. 133, 194106 (2010).
Whitfield, J. D., Biamonte, J. & Aspuru-Guzik, A. Simulation of electronic structure Hamiltonians using quantum computers. Mol. Phys. 109, 735–750 (2011).
Li, Z. et al. Solving quantum ground-state problems with nuclear magnetic resonance. Sci. Rep. 1, 88 (2011).
Griffiths, R. B. & Niu, C-S. Semiclassical Fourier transform for quantum computation. Phys. Rev. Lett. 76, 3228–3231 (1996).
Beckman, D., Chari, A. N., Devabhaktuni, S. & Preskill, J. Efficient networks for quantum factoring. Phys. Rev. A 54, 1034–1063 (1996).
Prevedel, R. et al. High-speed linear optics quantum computing using active feed-forward. Nature 445, 65–69 (2007).
Zhou, X-Q. et al. Adding control to arbitrary unknown quantum operations. Nature. Commun. 2, 413 (2011).
Ralph, T. C., Langford, N. K., Bell, T. B. & White, A. G. Linear optical controlled-NOT gate in the coincidence basis. Phys. Rev. A 65, 062324 (2001).
Hofmann, H. F. & Takeuchi, S. Quantum phase gate for photonic qubits using only beam splitters and postselection. Phys. Rev. A 66, 024308 (2001).
O'Brien, J. L., Pryde, G. J., White, A. G., Ralph, T. C. & Branning, D. Demonstration of an all-optical quantum controlled-NOT gate. Nature 426, 264–267 (2003).
O'Brien, J. L. et al. Quantum process tomography of a controlled-NOT gate. Phys. Rev. Lett. 93, 080502 (2004).
Kwiat, P. G., Waks, E., White, A. G., Appelbaum, I. & Eberhard, P. H. Ultrabright source of polarization-entangled photons. Phys. Rev. A 60, R773–R776 (1999).
Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46–52 (2001).
The authors thank S. Bartlett, R. Jozsa, G. McConnell, T. Ralph, T. Rudolph and P. Shadbolt for helpful discussions. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC), the European Research Council (ERC), PHORBITECH and the Centre for Nanoscience and Quantum Information (NSQI). J.O.B. acknowledges a Royal Society Wolfson Merit Award.
The authors declare no competing financial interests.
About this article
Cite this article
Martín-López, E., Laing, A., Lawson, T. et al. Experimental realization of Shor's quantum factoring algorithm using qubit recycling. Nature Photon 6, 773–776 (2012). https://doi.org/10.1038/nphoton.2012.259
This article is cited by
Nature Communications (2023)
The Journal of Supercomputing (2023)
npj Quantum Information (2022)
Scientific Reports (2022)
Nature Communications (2022)